Question

A random sample of size n = 55 is taken from a population with mean μ = −10.5 and standard deviation σ = 2. [You may find it useful to reference the z table.]

a. Calculate the expected value and the standard error for the sampling distribution of the sample mean. (Negative values should be indicated by a minus sign. Round "expected value" to 1 decimal place and "standard error" to 4 decimal places.)

b. What is the probability that the sample mean is less than −11? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What is the probability that the sample mean falls between −11 and −10? (Do not round intermediate calculations. Round "z" value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

Solution :

Given that,

mean = = −10.5

standard deviation = = −10.5

n = 55

= −10.5

The expected value

  =  (/n) = ( 2 / 55 ) = 0.2697

The standard error = 0.2697

P (   < 95 )

P (−11< x < −10)

P ( −11 −10.5 / 0.2697) < ( x -  / ) < ( −10 - −10.5 / 0.2697)

P ( 0.5 / 0.2697< z < - 0.5/ 0.2697)

P ( 1.85< z < - 1.85 )

P ( z < 0.97 ) - P ( z < - 0.97)

Using z table

= 0.8340 - 0.1660

= 0.6680

Probability = 0.6680